How can I rig a cube in Blender so that it bends and twists?
Ok, Here are the simple steps.

Create a Simple Cube and Subdivide it twice. (A> Select all, W> Subdivide)


Scale it 2 times (S2)
and Add a Single Bone (ShiftA> Armature > Single Bone).
Rename it to Root in (3D view > Properties region (N)> Item).

In edit Mode add 8 child bones (ShiftA) to root with tips facing the corners of the cube.
Turn on snapping (hold Ctrl) to precisely move them to corners. Create one bone first, snap and place at a corner, then simply duplicate (ShiftD) and rotate (R). Don't forget to select rootbone as parent before duplicating.

In object mode, parent your cube object to the armature (CtrlP). You are ready to animate. All corners work. Add more subdivisions for fine movement.